COMPOUND HARMONIC MOTION. 23 Hence pq will at every instant be the step from its mean position to a point which is moving in a simple harmonic motion of amplitude ac, period 27.oa. When therefore the figure is unrolled from the cylinder, the wavy curve (called the harmonic curve, or curve of sines because the ordinate pq is equal to ac. sin, that is, proportional 2 b'a to the sine of a multiple of the abscissa b'p) is the curve of positions of the simple harmonic motion aforesaid. The amplitude is the height of a wave, ac. The period is the length of a wave, b'b', every centimeter in that length representing a second of time. The curves of position of motions compounded of simple harmonic motions in one line may be constructed by actually compounding the curves of position of the several motions—that is, by adding together their ordinates to form the ordinate of the compound curve. Thus in m the figures the height of the dark curve above the horizontal line is at every point half the algebraic sum (which is more convenient for drawing than the whole sum) of the heights of the other two; as for example 2mq=mp + mr. A depth below the line is counted as a negative height. The first figure represents the composition of two simple harmonic motions of the same period; the second two such motions in which the period of one is half that of the other. The student should construct a series of these for different epochs of one of the motions, and then compare them with those figured in Thomson and Tait's Natural Philosophy, p. 43. In the case where the component motions have the same period, the resultant is a simple harmonic motion of that period. This follows at once from the corresponding theorem in regard to circular motions. Completing the parallelogram opqr, and drawing perpendiculars pl, qm, rn then a P=P1+P2 = ccos (nt + €), provided that c2= a2+b2+2ab cos (e̟ ̧ + €), and a sin €1 + b sin tan € = ၆၇ a cos e + b cos e any number It follows at once that the theorem is true for A The use of the jointed parallelogram opqr for compounding harmonic motions of different periods is exemplified in Sir W. Thomson's Tidal Clock. The clock has two hands whose lengths are proportional to the solar and lunar tides respectively, while their periods of revolution are equal to the periods of those tides. jointed parallelogram is constructed, having the hands of the clock for two sides. If the clock is properly set, the height of that extremity of the parallelogram which is furthest from the centre will be continually proportional to the height of the compound tide. For this purpose a series of horizontal lines at equal distances is drawn across the face of the clock, and the height is read off by running the eye along these to a vertical scale of feet in the middle. ON PROJECTION. The foot of the perpendicular from a point on a straight line or plane is called the orthogonal projection of that point on the line or plane, or more shortly (when no mistake can occur) the projection of the point. Thus the point m is the projection of p on the straight line aa'. We say also, by a natural extension, that the motion of m is the projection of the motion of p. Thus simple harmonic motion is the orthogonal projection of uniform circular motion on any straight line in the plane of the circle. When all the points of a figure are projected, the figure formed by their projections is called the projection of the original figure. Thus, for example (first figure of p. 22), the circle aba'b' is the projection of the ellipse cbc'b', for it is produced by drawing perpendiculars from every point of the ellipse to the plane. The point a is the projection of c, a' of c', p of q, etc.; b and b' are their own projections, being already in the plane of the circle. Instead of drawing lines perpendicular to a plane from all the points of a figure, we may also project it by drawing lines all parallel to one another, but in some other direction. This is called oblique projection, The ellipse cbc'b' is an oblique projection of the circle aba'b', for the lines ac, a'c', pq are all parallel to one another, although they are not at right angles to the plane of the ellipse. Orthogonal and oblique projections are both included under the name parallel projection, because in both cases the projection is made by drawing lines which are all parallel to one another. C b We may also project a figure on to a given plane by means of lines drawn through a fixed point; this is called central projection. It occurs whenever a shadow is cast by a luminous point. If we suppose the centre of projection c to move away to an infinite distance, the lines converging to it will all become parallel. Thus we see that parallel projection is only a particular case of central projection in which the centre of projection has gone away to an infinite distance. The shadow cast by a bright star is for all practical purposes a parallel projection. The projection of a straight line is made by drawing a plane through it and through the centre of projection. Thus if we draw the plane cab and produce it to meet the plane of projection in a'b', this line a'b' will be the projection of ab. In parallel projection we must draw through the line a plane parallel to the projecting lines, like the plane ab a'b' in the second figure. We see in this way that the projection of a straight line is always a straight line, and that, since the line and its projection are in the same plane, they must either meet at a finite distance or be parallel (meet at an infinite distance). In parallel projection, parallel lines are projected into parallel lines, and the ratio of their lengths is unaltered. Through the parallel lines ab, cd we must draw the planes aba'b', cdc'd' both parallel to the projecting lines, and therefore parallel to each other. These planes will consequently be cut by the plane of projection in the parallel lines a'b', c'd'. Moreover the triangles pbb', qdd, having their respective a sides parallel, are similar; therefore pb : qd=pb′ : qd', and so also ab: cd=a'b' : c'd'. The orthogonal projection of a finite straight line on a straight line or plane is equal in length to the length of the projected line multiplied by the cosine of its inclination to the straight line or plane. If pq is the projection of PQ, draw pq equal and parallel to PQ. Then Q7 is parallel to Pp and therefore perpendicular to pq; therefore the plane Qqq' is perpendicular to pq, and therefore q'q is perpendicular to pq. Hence pq=pq cos qpq=PQ x cosine of angle between PQ and pq. The orthogonal projection of an area on a plane is equal to the area multiplied by the cosine of its inclination to the PROJECTION OF AN AREA. A A B D 27 plane. This is clearly true for a rectangle ABCD, one of whose sides is parallel to the line of intersection of the planes. For the side AB is unaltered, and the other, BC, is altered into Bc, which is BC cos 0. Hence it is true for any area which can be made up of such rectangles. But any area A can be divided into such rectangles together with pieces over, by drawing lines across it at equal distances perpendicular to the intersection of the two planes, and then lines parallel to the intersection through the points when they meet the boundary. All these pieces over, taken together, are less than twice the strip whose height PQ is the difference in height between the lowest and highest point of the area; for those on either side of it can be slid sideways into that strip so as not to fill it. And by increasing the number of strips, and diminishing their breadth, we can make this as small as we like. Let then A' be the sum of the rectangles, then A' can be made to differ from A as little as we like. Now the projection of A' is A' cos 0, and this can be made to differ from the projection of A as little as we like. Therefore there can be no finite difference between the projection of A and A cos 0, because A' cos 0 can be made to differ as little as we like from both of them. PROPERTIES OF THE ELLIPSE. The ellipse may be defined in various ways, but for our purposes it is most convenient to define it as the parallel projection of a circle. This definition leads most easily to those properties of the curve which are chiefly useful in dynamic. Centre. The centre of a circle bisects every chord passing through it; such a chord is called a diameter. |